Theorem fococnv2

Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015)

	Ref	Expression
Assertion	fococnv2	|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )

Proof

Step	Hyp	Ref	Expression
1		fofun	|- ( F : A -onto-> B -> Fun F )
2		funcocnv2	|- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
3	1 2	syl	|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` ran F ) )
4		forn	|- ( F : A -onto-> B -> ran F = B )
5	4	reseq2d	|- ( F : A -onto-> B -> ( _I |` ran F ) = ( _I |` B ) )
6	3 5	eqtrd	|- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) )